Research Of Deep Learning On 3D Rotation Groups Based On Quaternion Algebra

3D data has a broad range of applications such as autonomous driving,robotics and gaming.Compared with 1D signal and 2D image,3D data usually presents a lack of regular spatial structure and has a higher requirement for rotation robustness.Based on these two considerations,many recent works proposed deep learning methods more suitable for 3D data and have improved the performance of 3D data analysis by a large margin.3D rotation is an important 3D data representation which express the relative relationship in 3D space more directly such as the joint rotation of human skeleton,the pose of objects in 3D space,etc.Compared with point cloud data,3D rotation has a unique group structure and is in non-Euclidean space.The algebraic operations used in ordinary real-valued networks are not closed to3 D rotation group,so the relative relationship of the 3D rotation data cannot be effectively extracted.Quaternion algebra provides a compact representation of 3D rotations without singularities.Based on unique mathematical properties of 3D rotation groups,we propose a novel quaternion product unit(QPU)to represent data on 3D rotation groups.The QPU leverages quaternion algebra and the law of 3D rotation group,representing 3D rotation data as quaternions and merging them via a weighted chain of Hamilton products.We prove that the representations derived by the proposed QPU can be disentangled into “rotation-invariant” features and “rotation-equivariant“ features,respectively,which supports the rationality and the efficiency of the QPU in theory.We design quaternion neural networks based on our QPUs and make our models compatible with existing deep learning models.Experiments on both synthetic and real-world data show that the proposed QPU is beneficial for the learning tasks requiring rotation robustness.